DOCSLIB.ORG

Abstracts: Plenary Speakers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Abstracts: Plenary Speakers Abstracts: Plenary Speakers David Blecher (Houston) [email protected] Applications of a kind of positivity in operator algebras Monday, August 8, MC4020, 2:00–2:50 We give some applications of a kind of positivity in nonselfadjoint operator algebras, related to new properties of approximate identities in operator al- gebras due to C. Read, and Read and the author. Much of this is joint work with Read or with M. Neal. H. Garth Dales (Leeds & Lancaster) [email protected] Multi-norms, duality, and the injectivity of Lp(G) Friday, August 5, MC4020, 9:00–9:50 In work with M. Polyakov [1], I studied when various left modules over a group algebra L1(G) are projective or injective or flat, and quite a few results were obtained. For example, it was noted that Lp(G) is injective whenever the locally compact group G is amenable, and it was conjectured that the converse holds. An attack on this conjecture led to its reformulation as a question involving ‘multi-norms’; I have spoken previously about multi- norms. In the present talk, I shall first recall quickly some notions from multi- norm theory - and introduce some new multi-norms. Connections between multi-norms are determined by using some theory of absolutely summing operators [4]. One new multi-norm is defined on Banach lattices, and resolves some questions in that theory. Second, I shall explain how this theory does indeed lead to a positive solution of the above conjecture (this is joint work with Matt Daws, Hung Le Pham, and Paul Ramsden [3]); some combinatorial conditions related to the amenability of groups from [5] will be mentioned. Finally I shall develop a new theory of ‘dual multi-norms’; this involves various notions of decomposability of Banach spaces in terms of multi-norms. The theory is particularly successful in the case of Banach lattices, mainly because of a striking theorem of the the late Nigel Kalton [6] that I shall explain. I hope that all relevant papers will be placed in a special part of my website by the time that the conference begins. 1 [1] H. G. Dales and M. E. Polyakov, Homological properties of modules over group algebras, Proc. London Math. Soc. (3), 89 (2004), 390–426. [2] H. G. Dales and M. E. Polyakov, Multi-normed spaces, preprint. [3] H. G. Dales, M. Daws, H. L. Pham, and P. Ramsden, Multi-norms and the injectivity of Lp(G), submitted to Proc. London Math. Soc., arXiv:1101.4320v1 [math.FA]. [4] H. G. Dales, M. Daws, H. L. Pham, and P. Ramsden, Equivalence of multi-norms, in preparation. [5] H. G. Dales, M. Daws, H. L. Pham, and P. Ramsden, Multi-norms and the amenability of groups, in preparation. [6] N. J. Kalton, Hermitian operators on complex Banach lattices and a problem of Garth Dales, submitted to Proc. London Math. Soc. Matthew Daws (Leeds) [email protected] Shift invariant preduals of group algebras Saturday, August 6, MC4020, 9:00–9:50 For a discrete semigroup S, the Banach space ℓ1(S) admits a canonical pre- dual c0(S). When, say, S is cancellative (in particular, if S is a group) then the resulting weak∗-topology on ℓ1(S) is such that the algebra product is separately continuous. We say that ℓ1(S) is a dual Banach algebra, with respect to c0(S). Now, as a Banach space, ℓ1(S) admits many different preduals, which induce different weak∗-topologies upon ℓ1(S). However, these might not make ℓ1(S) into a dual Banach algebra. In the first part of the talk, I will discuss joint work with Hung le Pham and Stuart White. We will exhibit semigroups S such that c0(S) is the only predual turning ℓ1(S) into a dual Banach algebra. The algebra ℓ1(S) is also a coalgebra for a coproduct which pre-dualises the multiplication on ℓ∞(S). If S is a group, we show that c0(S) is the only predual making both the product and the coproduct weak∗-continuous. However, we give examples of semigroups S which admit continuum many different preduals, all making the product and coproduct weak∗-continuous on ℓ1(S). In the second part of the talk, I will discuss joint work with Richard Haydon, Thomas Schlumprecht and Stuart White. Here we find a relatively concrete construction of a one-parameter family of preduals of ℓ1(Z), each turning ℓ1(Z) into a dual Banach algebra. Time allowing, I will sketch some 2 abstract theory which “classifies” all dual Banach algebra preduals of ℓ1(G) using semigroup compactification theory. In the case of Z, this allows for a more abstract construction of preduals with many interesting properties (in particular, we find a dual Banach algebra predual which is not isomorphic to c0 just as a Banach space). Jean Erstele (Bordeaux I) [email protected] Lower norm estimates for functional calculus on quasinilpotent semigroups and pseudospectrum of generators Thursday, August 4, MC4061, 9:00-9:50 We consider here semigroups (T (t))=(e−tA) of bounded linear operators on a Banach space X which norm-continuous on (0, +∞), or analytic on a sector Sα := {z ∈ C \{0}||arg(z)| < α} for some α ∈ (0,π/2], and look for lower estimates for the distance kT (t)−T (s)k, or more generally for kf(A)k, where A is the infinitesimal generator of the semigroup and f a sum of suitable exponential-polynomial functions. We focus on the case of quasinilpotent semigroups, and propose a new approach based on the pseudospectrum of their infinitesimal generator. Zhiguo Hu (Windsor) [email protected] A Stone-von Neumann theorem on quantum groups and the convolution algebra (T (L2(G)),⊲) Wednesday, August 10, MC4061, 11:30-12:20 For a locally compact quantum group G, the space T (L2(G)) of trace class operators on L2(G) is a Banach algebra with the multiplication ⊲ induced by the right fundamental unitary of G. In this talk, we present a general- ized Stone-von Neumann theorem for locally compact quantum groups via the convolution ⊲, extending the classical Stone-von Neumann theorem on locally compact groups. We discuss applications of this theorem to various problems around the quantum group algebra L1(G) and the operator convo- lution algebra (T (L2(G)),⊲). This is joint work with Matthias Neufang and Zhong-Jin Ruan. 3 Matthias Neufang (Carleton & Fields & Lille I) [email protected] On problems of Ghahramani–Lau and Johnson Friday, August 5, MC4020, 2:00–2:50 We present the solution, in full generality, of the Ghahramani–Lau conjecture (1994) and the solution, for a large class of Banach spaces, of a problem raised by B. Johnson (1972); the first is joint work with V. Losert, J. Pachl and J. Stepr¯ans, the second with my Ph.D. student D. Poulin. The conjecture of F. Ghahramani and A.T.-M. Lau states that the mea- sure algebra M(G) over any locally compact group G is strongly Arens irreg- ular. A key ingredient in our proof is a factorization result we obtain for the canonical module actions of M(G)∗∗ on M(G)∗. We also discuss topological centres of the biduals of some natural (one-sided) ideals in M(G), as well as of measure algebras over non locally compact groups. Moreover, we point out how a version of the above-mentioned factorization technique can be used to study equi uniform continuity over locally compact quantum groups; this is joint work with J. Pachl and P. Salmi. The problem posed by B. Johnson in which we shall be interested, con- cerns the (non-)amenability of the algebra B(E) of bounded linear operators on a Banach space E. We prove that for any Banach space X which is not complemented in its bidual, B(X∗∗) is non-amenable, thus generalizing the previously known case X = c0 due to N. Ozawa. In particular, B(M) is non- amenable for any (infinite-dimensional) atomic von Neumann algebra M. We also show that for an arbitrary (infinite-dimensional) von Neumann algebra M with separable predual, B(M) does not have a countable approximate di- agonal. We obtain analogous results in the category of operator spaces and completely bounded maps. Narutaka Ozawa (RIMS Kyoto) [email protected] Survey on weak amenability for groups Monday, August 8, MC4020, 9:00–9:50 One of the most basic tool in the study of Fourier series is Fej´er’s theorem which states that the Ces`aro means of partial sums of the Fourier series of a given function converge to that function. More precisely, the Ces`aro mean |k| functions φn(k) = (1 − n ) ∨ 0 on Z are uniformly bounded as Herz–Schur 4 multipliers (in fact positive and contractive). A group Γ is said to be weakly amenable or have the Cowling–Haagerup property if Fej´er’s theorem holds for Γ; namely if there is a sequence (φn) of finitely supported functions with uniformly bounded Herz–Schur norm which converges to 1. (It is amenable if φn’s are not only bounded but moreover positive type.) The class of weakly amenable groups contains many interesting examples such as free groups, which are not amenable. I will give a survey on weakly amenable groups. Warning: The term “weakly amenable” for groups is unrelated to the same term for Banach algebras (so far). Vern Paulsen (Houston) [email protected] A multivariable analogue of Ando’s theorem on numerical radius and C*-algebras with WEP Tuesday, August 9, MC2065, 2:00–2:50 A classic theorem of T.
Load more

Recommended publications
  • Differentiation of Operator Functions in Non-Commutative Lp-Spaces B
    ARTICLE IN PRESS Journal of Functional Analysis 212 (2004) 28–75 Differentiation of operator functions in non-commutative Lp-spaces B. de Pagtera and F.A. Sukochevb,Ã a Department of Mathematics, Faculty ITS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands b School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042 SA, Australia Received 14 May 2003; revised 18 September 2003; accepted 15 October 2003 Communicated by G. Pisier Abstract The principal results in this paper are concerned with the description of differentiable operator functions in the non-commutative Lp-spaces, 1ppoN; associated with semifinite von Neumann algebras. For example, it is established that if f : R-R is a Lipschitz function, then the operator function f is Gaˆ teaux differentiable in L2ðM; tÞ for any semifinite von Neumann algebra M if and only if it has a continuous derivative. Furthermore, if f : R-R has a continuous derivative which is of bounded variation, then the operator function f is Gaˆ teaux differentiable in any LpðM; tÞ; 1opoN: r 2003 Elsevier Inc. All rights reserved. 1. Introduction Given an arbitrary semifinite von Neumann algebra M on the Hilbert space H; we associate with any Borel function f : R-R; via the usual functional calculus, the corresponding operator function a/f ðaÞ having as domain the set of all (possibly unbounded) self-adjoint operators a on H which are affiliated with M: This paper is concerned with the study of the differentiability properties of such operator functions f : Let LpðM; tÞ; with 1pppN; be the non-commutative Lp-space associated with ðM; tÞ; where t is a faithful normal semifinite trace on the von f Neumann algebra M and let M be the space of all t-measurable operators affiliated ÃCorresponding author.
    [Show full text]
  • Introduction to Operator Spaces
    Lecture 1: Introduction to Operator Spaces Zhong-Jin Ruan at Leeds, Monday, 17 May , 2010 1 Operator Spaces A Natural Quantization of Banach Spaces 2 Banach Spaces A Banach space is a complete normed space (V/C, k · k). In Banach spaces, we consider Norms and Bounded Linear Maps. Classical Examples: ∗ C0(Ω),M(Ω) = C0(Ω) , `p(I),Lp(X, µ), 1 ≤ p ≤ ∞. 3 Hahn-Banach Theorem: Let V ⊆ W be Banach spaces. We have W ↑ & ϕ˜ ϕ V −−−→ C with kϕ˜k = kϕk. It follows from the Hahn-Banach theorem that for every Banach space (V, k · k) we can obtain an isometric inclusion (V, k · k) ,→ (`∞(I), k · k∞) ∗ ∗ where we may choose I = V1 to be the closed unit ball of V . So we can regard `∞(I) as the home space of Banach spaces. 4 Classical Theory Noncommutative Theory `∞(I) B(H) Banach Spaces Operator Spaces (V, k · k) ,→ `∞(I)(V, ??) ,→ B(H) norm closed subspaces of B(H)? 5 Matrix Norm and Concrete Operator Spaces [Arveson 1969] Let B(H) denote the space of all bounded linear operators on H. For each n ∈ N, n H = H ⊕ · · · ⊕ H = {[ξj]: ξj ∈ H} is again a Hilbert space. We may identify ∼ Mn(B(H)) = B(H ⊕ ... ⊕ H) by letting h i h i X Tij ξj = Ti,jξj , j and thus obtain an operator norm k · kn on Mn(B(H)). A concrete operator space is norm closed subspace V of B(H) together with the canonical operator matrix norm k · kn on each matrix space Mn(V ).
    [Show full text]
  • On the Space of Generalized Fluxes for Loop Quantum Gravity
    Home Search Collections Journals About Contact us My IOPscience On the space of generalized fluxes for loop quantum gravity This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2013 Class. Quantum Grav. 30 055008 (http://iopscience.iop.org/0264-9381/30/5/055008) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 194.94.224.254 The article was downloaded on 07/03/2013 at 11:02 Please note that terms and conditions apply. IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 30 (2013) 055008 (24pp) doi:10.1088/0264-9381/30/5/055008 On the space of generalized fluxes for loop quantum gravity B Dittrich1,2, C Guedes1 and D Oriti1 1 Max Planck Institute for Gravitational Physics, Am Muhlenberg¨ 1, D-14476 Golm, Germany 2 Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo ON N2 L 2Y5, Canada E-mail: [email protected], [email protected] and [email protected] Received 5 September 2012, in final form 13 January 2013 Published 5 February 2013 Online at stacks.iop.org/CQG/30/055008 Abstract We show that the space of generalized fluxes—momentum space—for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections—position space— due to the non-Abelianess of the gauge group SU(2). From the Abelianization of SU(2),U(1)3, we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse graining of the underlying graphs.
    [Show full text]
  • Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
    UNIVERSITY of CALIFORNIA Santa Barbara Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering by Ryan M. Mohr Committee in charge: Professor Igor Mezi´c,Chair Professor Bassam Bamieh Professor Jeffrey Moehlis Professor Mihai Putinar June 2014 The dissertation of Ryan M. Mohr is approved: Bassam Bamieh Jeffrey Moehlis Mihai Putinar Igor Mezi´c,Committee Chair May 2014 Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems Copyright c 2014 by Ryan M. Mohr iii To my family and dear friends iv Acknowledgements Above all, I would like to thank my family, my parents Beth and Jim, and brothers Brennan and Gralan. Without their support, encouragement, confidence and love, I would not be the person I am today. I am also grateful to my advisor, Professor Igor Mezi´c,who introduced and guided me through the field of dynamical systems and encouraged my mathematical pursuits, even when I fell down the (many) proverbial (mathematical) rabbit holes. I learned many fascinating things from these wandering forays on a wide range of topics, both contributing to my dissertation and not. Our many interesting discussions on math- ematics, research, and philosophy are among the highlights of my graduate school career. His confidence in my abilities and research has been a constant source of encouragement, especially when I felt uncertain in them myself. His enthusiasm for science and deep, meaningful work has set the tone for the rest of my career and taught me the level of research problems I should consider and the quality I should strive for.
    [Show full text]
  • Chapter 2 C -Algebras
    Chapter 2 C∗-algebras This chapter is mainly based on the first chapters of the book [Mur90]. Material bor- rowed from other references will be specified. 2.1 Banach algebras Definition 2.1.1. A Banach algebra C is a complex vector space endowed with an associative multiplication and with a norm k · k which satisfy for any A; B; C 2 C and α 2 C (i) (αA)B = α(AB) = A(αB), (ii) A(B + C) = AB + AC and (A + B)C = AC + BC, (iii) kABk ≤ kAkkBk (submultiplicativity) (iv) C is complete with the norm k · k. One says that C is abelian or commutative if AB = BA for all A; B 2 C . One also says that C is unital if 1 2 C , i.e. if there exists an element 1 2 C with k1k = 1 such that 1B = B = B1 for all B 2 C . A subalgebra J of C is a vector subspace which is stable for the multiplication. If J is norm closed, it is a Banach algebra in itself. Examples 2.1.2. (i) C, Mn(C), B(H), K (H) are Banach algebras, where Mn(C) denotes the set of n × n-matrices over C. All except K (H) are unital, and K (H) is unital if H is finite dimensional. (ii) If Ω is a locally compact topological space, C0(Ω) and Cb(Ω) are abelian Banach algebras, where Cb(Ω) denotes the set of all bounded and continuous complex func- tions from Ω to C, and C0(Ω) denotes the subset of Cb(Ω) of functions f which vanish at infinity, i.e.
    [Show full text]
  • Operator Spaces
    Operator Spaces. Alonso Delf´ın University of Oregon. November 19, 2020 Abstract The field of operator spaces is an important branch of functional analysis, commonly used to generalize techniques from Banach space theory to algebras of operators on Hilbert spaces. A \concrete" operator space E is a closed subspace of L(H) for a Hilbert space H. Almost 35 years ago, Zhong-Jin Ruan gave an abstract characterization of operator spaces, which allows us to forget about the concrete Hilbert space H. Roughly speaking, an \abstract" operator space consists of a normed space E together with a family of matrix norms on Mn(E) satisfying two axioms. Ruan's Theorem states that any abstract operator space is completely isomorphic to a concrete one. On this document I will give a basic introduction to operator spaces, providing many examples. I will discuss completely bounded maps (the morphisms in the category of operator spaces) and try to give an idea of why abstract operator spaces are in fact concrete ones. Time permitting, I will explain how operator spaces are used to define a non-selfadjoint version of Hilbert modules and I'll say how this might be useful for my research. 1 Definitions and Examples Definition 1.1. Let H be a Hilbert space. An operator space E is a closed subspace of L(H). Example 1.2. 1. Any C∗-algebra A is an operator space. 2. Let H1 and H2 be Hilbert spaces. Then L(H1; H2) is regarded as an operator space by identifying L(H1; H2) in L(H1 ⊕ H2) by 0 0 L(H ; H ) 3 a 7! 2 L(H ⊕ H ) 1 2 a 0 1 2 3.
    [Show full text]
  • Banach Algebras
    Banach Algebras Yurii Khomskii Bachelor Thesis Department of Mathematics, Leiden University Supervisor: Dr. Marcel de Jeu April 18, 2005 i Contents Foreword iv 1. Algebraic Concepts 1 1.1. Preliminaries . 1 1.2. Regular Ideals . 3 1.3. Adjoining an Identity . 4 1.4. Quasi-inverses . 8 2. Banach Algebras 10 2.1. Preliminaries of Normed and Banach Algebras . 10 2.2. Inversion and Quasi-inversion in Banach Algebras . 14 3. Spectra 18 3.1. Preliminaries . 18 3.2. Polynomial Spectral Mapping Theorem and the Spectral Radius Formula . 22 4. Gelfand Representation Theory 25 4.1. Multiplicative Linear Functionals and the Maximal Ideal Space . 25 4.2. The Gelfand Topology . 30 4.3. The Gelfand Representation . 31 4.4. The Radical and Semi-simplicity . 33 4.5. Generators of Banach algebras . 34 5. Examples of Gelfand Representations 36 5.1. C (X ) for X compact and Hausdorff . 36 5.2. C 0(X ) for X locally compact and Hausdorff. 41 5.3. Stone-Cecˇ h compactification . 42 5.4. A(D) . 44 5.5. AC (Γ) . 46 5.6. H 1 . 47 ii iii Foreword The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard exam- ple was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infin- ity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications.
    [Show full text]
  • Mathematical Tripos Part III Lecture Courses in 2013-2014
    Mathematical Tripos Part III Lecture Courses in 2013-2014 Department of Pure Mathematics & Mathematical Statistics Department of Applied Mathematics & Theoretical Physics Notes and Disclaimers. • Students may take any combination of lectures that is allowed by the timetable. The examination timetable corresponds to the lecture timetable and it is therefore not possible to take two courses for examination that are lectured in the same timetable slot. There is no requirement that students study only courses offered by one Department. • The code in parentheses after each course name indicates the term of the course (M: Michaelmas; L: Lent; E: Easter), and the number of lectures in the course. Unless indicated otherwise, a 16 lecture course is equivalent to 2 credit units, while a 24 lecture course is equivalent to 3 credit units. Please note that certain courses are non-examinable, and are indicated as such after the title. Some of these courses may be the basis for Part III essays. • At the start of some sections there is a paragraph indicating the desirable previous knowledge for courses in that section. On one hand, such paragraphs are not exhaustive, whilst on the other, not all courses require all the pre-requisite material indicated. However you are strongly recommended to read up on the material with which you are unfamiliar if you intend to take a significant number of courses from a particular section. • The courses described in this document apply only for the academic year 2013-14. Details for subsequent years are often broadly similar, but not necessarily identical. The courses evolve from year to year.
    [Show full text]
  • Noncommutative Functional Analysis for Undergraduates"
    N∞M∞T Lecture Course Canisius College Noncommutative functional analysis David P. Blecher October 22, 2004 2 Chapter 1 Preliminaries: Matrices = operators 1.1 Introduction Functional analysis is one of the big fields in mathematics. It was developed throughout the 20th century, and has several major strands. Some of the biggest are: “Normed vector spaces” “Operator theory” “Operator algebras” We’ll talk about these in more detail later, but let me give a micro-summary. Normed (vector) spaces were developed most notably by the mathematician Banach, who not very subtly called them (B)-spaces. They form a very general framework and tools to attack a wide range of problems: in fact all a normed (vector) space is, is a vector space X on which is defined a measure of the ‘length’ of each ‘vector’ (element of X). They have a huge theory. Operator theory and operator algebras grew partly out of the beginnings of the subject of quantum mechanics. In operator theory, you prove important things about ‘linear functions’ (also known as operators) T : X → X, where X is a normed space (indeed usually a Hilbert space (defined below). Such operators can be thought of as matrices, as we will explain soon. Operator algebras are certain collections of operators, and they can loosely be thought of as ‘noncommutative number fields’. They fall beautifully within the trend in mathematics towards the ‘noncommutative’, linked to discovery in quantum physics that we live in a ‘noncommutative world’. You can study a lot of ‘noncommutative mathematics’ in terms of operator algebras. The three topics above are functional analysis.
    [Show full text]
  • Spectral Theory
    SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensional case. Namely, we first show that an eigenvector exists, and then we show that there is an orthonormal basis of eigenvectors by an inductive argument (in the form of an application of Zorn's lemma). We first recall a few facts about self-adjoint operators. Proposition 1.1. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. (1) If W ⊂ V is a T -invariant subspace, then W ? is T -invariant. (2) For all v 2 V , hT v; vi 2 R. (3) Let Vλ = fv 2 V j T v = λvg. Then Vλ ? Vµ whenever λ 6= µ. We will need one further technical fact that characterizes the norm of a self-adjoint operator. Lemma 1.2. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. Then kT k = supfjhT v; vij j kvk = 1g. Proof. Let α = supfjhT v; vij j kvk = 1g. Evidently, α ≤ kT k. We need to show the other direction. Given v 2 V with T v 6= 0, setting w0 = T v=kT vk gives jhT v; w0ij = kT vk.
    [Show full text]
  • MAT 449 : Representation Theory
    MAT 449 : Representation theory Sophie Morel December 17, 2018 Contents I Representations of topological groups5 I.1 Topological groups . .5 I.2 Haar measures . .9 I.3 Representations . 17 I.3.1 Continuous representations . 17 I.3.2 Unitary representations . 20 I.3.3 Cyclic representations . 24 I.3.4 Schur’s lemma . 25 I.3.5 Finite-dimensional representations . 27 I.4 The convolution product and the group algebra . 28 I.4.1 Convolution on L1(G) and the group algebra of G ............ 28 I.4.2 Representations of G vs representations of L1(G) ............. 33 I.4.3 Convolution on other Lp spaces . 39 II Some Gelfand theory 43 II.1 Banach algebras . 43 II.1.1 Spectrum of an element . 43 II.1.2 The Gelfand-Mazur theorem . 47 II.2 Spectrum of a Banach algebra . 48 II.3 C∗-algebras and the Gelfand-Naimark theorem . 52 II.4 The spectral theorem . 55 III The Gelfand-Raikov theorem 59 III.1 L1(G) ....................................... 59 III.2 Functions of positive type . 59 III.3 Functions of positive type and irreducible representations . 65 III.4 The convex set P1 ................................. 68 III.5 The Gelfand-Raikov theorem . 73 IV The Peter-Weyl theorem 75 IV.1 Compact operators . 75 IV.2 Semisimplicity of unitary representations of compact groups . 77 IV.3 Matrix coefficients . 80 IV.4 The Peter-Weyl theorem . 86 IV.5 Characters . 87 3 Contents IV.6 The Fourier transform . 90 IV.7 Characters and Fourier transforms . 93 V Gelfand pairs 97 V.1 Invariant and bi-invariant functions .
    [Show full text]
  • The Maximal Operator Space of a Normed Space
    Proceedings of the Edinburgh Mathematical Society (1996) 39, 309-323 .<* THE MAXIMAL OPERATOR SPACE OF A NORMED SPACE by VERN I. PAULSEN* (Received 27th June 1994) We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X),MAX(y)) = rj(A', Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of "free" unitaries. 1991 Mathematics subject classification: 46L05, 46M05. 1. Introduction Given any (complex) vector space X, let Mn(X) denote the vector space of nxn matrices with entries from X. If B(H) denotes the bounded linear operators on a Hilbert space H then Mn(B(H)) is endowed with a natural norm via the identification, Mn{B(H)) = B(H®•••®H) (n copies). If X is any subspace of B(H) then the inclusion Mn(X)sMn(B(H)) endows Mn(X) with a norm. These are generally called a matrix- norm on X. Subspaces of B(H) are called operator spaces to indicate that they are more than normed spaces, but normed spaces with a distinguished family of norms on the spaces, Mn(X). If X is only a normed space, then any linear isometry <p:X->B(H) for some H endows each Mn(X) with a norm, i.e. makes X an operator space. Generally, for a normed space X the norms on Mn(X) thus obtained are not unique.
    [Show full text]